It wasn’t until the early 1960’s that the next major advance was made. Independently, the mathematicians John Selfridge of Northern Illinois University and John Conway of Cambridge University came up with a protocol for dividing the cake between three people in a way that is both fair and envy-free. It goes like this...
Alice cuts the cake in a way that she considers fair.
Next, Bertha looks at the pieces and decides whether she thinks that one piece is larger than the rest, or whether two or more pieces are tied largest.
If Bertha considers that two or more pieces are tied largest, they choose pieces in the following order: Claire (who is happy because she chooses first), Bertha (who is happy because at least one of her tied largest pieces is left for her) and finally Alice (who is happy because she originally divided the cake).
If however Bertha considers that one piece is larger than the rest, she must trim that piece so that in her opinion it equals the second largest in size. Remember this piece - we’ll call it the "trimmed" piece - as it’s important later on. The leftovers we’ll put on one side for now.
Now they can again choose in the order Claire, Bertha, Alice, though if Claire doesn’t choose the trimmed piece then Bertha must, as Alice won’t consider it fair. Again, they are all happy and un-envious, for the same reasons as before.
Ok, so far so good. We’ve achieved what the mathematicians call a partial envy-free allocation of the cake, in the sense that we’ve divided most of the cake in an envy-free and fair manner. But what of the trimmings? Surely this process will repeat for ever?
This is where it gets quite subtle. We know that one of Claire and Bertha took an un-trimmed piece, and one took the trimmed piece. We’ll call the person who took the un-trimmed piece the "divider", and the other the "non-divider". The divider now divides the trimmings into three, and they choose in the order non-divider, Alice, divider.
Now then, the non-divider is happy and doesn’t envy anyone, because she chooses first. The divider is similarly happy because she divided the trimmings and is therefore happy with any of the pieces. Alice is not envious of the divider, as she chooses before her. The question is, why is Alice not envious of non-divider, who chooses first from the trimmings?
The answer lies in the fact that Alice originally divided the cake in a way she thought was fair. Bertha subsequently trimmed a piece, and the non-divider was defined as the one of Claire and Bertha who took that trimmed piece. Alice therefore has what the mathematicians call "an irrevocable advantage" over the non-divider, as she believes that even if the non-divider got all the trimmings, she would only have had as much as Alice got in the first round alone. So, Alice can never be envious of the non-divider and we have our solution.
Since the 60’s some further advances have been made in this area:
in 1992 a general solution was found for an arbitrary number of people, by Steven Brams and Alan Taylor in New York. However, this solution is not perfect as it isn’t finite but continues repeating a series of steps until there are only crumbs left and no-one is worried about them. Interestingly, the solution for four people starts by dividing the cake into five equal pieces!
Later, Brams and Taylor found a finite solution for the case of four people, though it’s fiendishly complex and involves 20 steps - with 14 of them repeated up to 11 times.
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